eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2016-03-01
5
1
1
14
10.22108/toc.2016.9513
9513
Skew Randi'c matrix and skew Randi'c energy
Ran Gu
guran323@163.com
1
Fei Huang
huangfei06@126.com
2
Xueliang Li
lxl@nankai.edu.cn
3
Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China
Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China
Center for Combinatorics, Nankai University, Tianjin 300071, China
Let $G$ be a simple graph with an orientation $sigma$, which assigns to each edge a direction so that $G^sigma$ becomes a directed graph. $G$ is said to be the underlying graph of the directed graph $G^sigma$. In this paper, we define a weighted skew adjacency matrix with Rand'c weight, the skew Randi'c matrix ${bf R_S}(G^sigma)$, of $G^sigma$ as the real skew symmetric matrix $[(r_s)_{ij}]$ where $(r_s)_{ij} = (d_id_j)^{-frac{1}{2}}$ and $(r_s)_{ji} = -(d_id_j)^{-frac{1}{2}}$ if $v_i rightarrow v_j$ is an arc of $G^sigma$, otherwise $(r_s)_{ij} = (r_s)_{ji} = 0$. We derive some properties of the skew Randi'c energy of an oriented graph. Most properties are similar to those for the skew energy of oriented graphs. But, surprisingly, the extremal oriented graphs with maximum or minimum skew Randi'c energy are completely different, no longer being some kinds of oriented regular graphs.
http://toc.ui.ac.ir/article_9513_5dd2d75be3009b50ce663bc68f39cb1e.pdf
oriented graph
skew Randi'c matrix
skew Randi'c energy
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2016-03-01
5
1
15
23
10.22108/toc.2016.9372
9372
Skew equienergetic digraphs
Harishchandra S. Ramane
hsramane@yahoo.com
1
K. Channegowda Nandeesh
nandeeshkc@yahoo.com
2
Ivan Gutman
gutman@kg.ac.rs
3
Xueliang Li
lxl@nankai.edu.cn
4
Karnatak University, Dharwad, India
Karnatak University, Dharwad
University of Kragujevac, 34000 Kragujevac
Nankai University, Tianjin
Let $D$ be a digraph with skew-adjacency matrix $S(D)$. The skew energy of $D$ is defined as the sum of the norms of all eigenvalues of $S(D)$. Two digraphs are said to be skew equienergetic if their skew energies are equal. We establish an expression for the characteristic polynomial of the skew adjacency matrix of the join of two digraphs, and for the respective skew energy, and thereby construct non-cospectral, skew equienergetic digraphs on $n$ vertices, for all $n geq 6$. Thus we arrive at the solution of some open problems proposed in [X. Li, H. Lian, A survey on the skew energy of oriented graphs, arXiv:1304.5707].
http://toc.ui.ac.ir/article_9372_1d921f94d58d62e8f06d43db2dc426d5.pdf
energy of graph
skew energy
skew equienergetic digraphs
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2016-03-01
5
1
25
35
10.22108/toc.2016.8594
8594
Weighted Szeged indices of some graph operations
Kannan Pattabiraman
pramank@gmail.com
1
P. Kandan
kandan2k@gmail.com
2
Annamalai University
Annamalai University
In this paper, the weighted Szeged indices of Cartesian product and Corona product of two connected graphs are obtained. Using the results obtained here, the weighted Szeged indices of the hypercube of dimension $n$, Hamming graph, $C_4$ nanotubes, nanotorus, grid, $t-$fold bristled, sunlet, fan, wheel, bottleneck graphs and some classes of bridge graphs are computed.
http://toc.ui.ac.ir/article_8594_7eec41f6c4504ac1095447614ce5721c.pdf
Graph products
Szeged index
weighted Szeged index
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2016-03-01
5
1
37
48
10.22108/toc.2016.7890
7890
ON $bullet$-LICT signed graohs $L_{bullet_c}(S)$ and $bullet$-LINE signed graohs $L_bullet(S)$
Mukti Acharya
mukti1948@gmail.com
1
Rashmi Jain
rashmi2011f@gmail.com
2
Sangita Kansal
sangita_kansal15@rediffmail.com
3
DELHI TECHNOLOGICAL UNIVERSITY,
DELHI - INDIA
DELHI TECHNOLOGICAL UNIVERSITY,
DELHI - INDIA
DELHI TECHNOLOGICAL UNIVERSITY,
DELHI - INDIA
A signed graph (or, in short, sigraph) $S=(S^u,sigma)$ consists of an underlying graph $S^u :=G=(V,E)$ and a function $sigma:E(S^u)longrightarrow {+,-}$, called the signature of $S$. A marking of $S$ is a function $mu:V(S)longrightarrow {+,-}$. The canonical marking of a signed graph $S$, denoted $mu_sigma$, is given as $$mu_sigma(v) := prod_{vwin E(S)}sigma(vw).$$ The line graph of a graph $G$, denoted $L(G)$, is the graph in which edges of $G$ are represented as vertices, two of these vertices are adjacent if the corresponding edges are adjacent in $G$. There are three notions of a line signed graph of a signed graph $S=(S^u,sigma)$ in the literature, viz., $L(S)$, $L_times(S)$ and $L_bullet(S)$, all of which have $L(S^u)$ as their underlying graph; only the rule to assign signs to the edges of $L(S^u)$ differ. Every edge $ee'$ in $L(S)$ is negative whenever both the adjacent edges $e$ and $e'$ in S are negative, an edge $ee'$ in $L_times(S)$ has the product $sigma(e)sigma(e')$ as its sign and an edge $ee'$ in $L_bullet(S)$ has $mu_sigma(v)$ as its sign, where $vin V(S)$ is a common vertex of edges $e$ and $e'$. The line-cut graph (or, in short, lict graph) of a graph $G=(V,E)$, denoted by $L_c(G)$, is the graph with vertex set $E(G)cup C(G)$, where $C(G)$ is the set of cut-vertices of $G$, in which two vertices are adjacent if and only if they correspond to adjacent edges of $G$ or one vertex corresponds to an edge $e$ of $G$ and the other vertex corresponds to a cut-vertex $c$ of $G$ such that $e$ is incident with $c$. In this paper, we introduce dot-lict signed graph (or $bullet$-lict signed graph} $L_{bullet_c}(S)$, which has $L_c(S^u)$ as its underlying graph. Every edge $uv$ in $L_{bullet_c}(S)$ has the sign $mu_sigma(p)$, if $u, v in E(S)$ and $pin V(S)$ is a common vertex of these edges, and it has the sign $mu_sigma(v)$, if $uin E(S)$ and $vin C(S)$. we characterize signed graphs on $K_p$, $pgeq2$, on cycle $C_n$ and on $K_{m,n}$ which are $bullet$-lict signed graphs or $bullet$-line signed graphs, characterize signed graphs $S$ so that $L_{bullet_c}(S)$ and $L_bullet(S)$ are balanced. We also establish the characterization of signed graphs $S$ for which $Ssim L_{bullet_c}(S)$, $Ssim L_bullet(S)$, $eta(S)sim L_{bullet_c}(S)$ and $eta(S)sim L_bullet(S)$, here $eta(S)$ is negation of $S$ and $sim$ stands for switching equivalence.
http://toc.ui.ac.ir/article_7890_ce0590d708f808d60b29942f13824a78.pdf
Signed graph
Balance
Switching
$bullet$-line signed graph
$bullet$-lict signed graph
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2016-03-01
5
1
49
55
10.22108/toc.2016.9956
9956
Ordering of trees by multiplicative second Zagreb index
Mehdi Eliasi
eliasi@math.iut.ac.ir
1
Ali Ghalavand
ali797ghalavand@gmail.com
2
Department of Mathematics and Computer Science , Faculty of Khansar, Khansar, Iran
Department of Mathematics and Computer Science, Faculty of Khansar, University of Isfahan, P.O.Box 87931133111, Khansar, Iran
For a graph $G$ with edge set $E(G)$, the multiplicative second Zagreb index of $G$ is defined as $Pi_2(G)=Pi_{uvin E(G)}[d_G(u)d_G(v)]$, where $d_G(v)$ is the degree of vertex $v$ in $G$. In this paper, we identify the eighth class of trees, with the first through eighth smallest multiplicative second Zagreb indeces among all trees of order $ngeq 14$.
http://toc.ui.ac.ir/article_9956_8efd28a3432a71695fc6a83d711c626e.pdf
multiplicative second Zagreb index
graph operation
tree